> Note that the probability of having some entanglement between your partial states goes up rather dramatically with the number of them that you have
Entanglement is not binary, it is continuous. If you start with states like CPHASE(5°)|TT>, a few rounds of distillation will have turned them into states like CPHASE(0.0000000000000001°)|TT>. Sure the output states are "still entangled", but the amount of entanglement is so negligible that you don't have to care. Such small distortions won't prevent trillion step computations from working.
> Note that the probability of having some entanglement between your partial states goes up rather dramatically with the number of them that you have
Entanglement is not binary, it is continuous. If you start with states like CPHASE(5°)|TT>, a few rounds of distillation will have turned them into states like CPHASE(0.0000000000000001°)|TT>. Sure the output states are "still entangled", but the amount of entanglement is so negligible that you don't have to care. Such small distortions won't prevent trillion step computations from working.